The Capital Asset Pricing Model, the Markowitz Portfolio Model - Coursework Example

MODERN PORTFOLIO THEORY ESSAY Introduction The Markowitz portfolio model is premised on the common observation that the average investor is risk-averse and would not expose his capital unless there is a premium for the additional risk he is taking. Risk is related to uncertainty about future outcomes. All other things being equal, an investor would prefer less risk rather than more risk for a given outcome; and for a given risk, he wants if possible to maximise his returns. (See Hagin 1978, Reilly and Brown 1997, Block and Hirt 2002). He can only be induced to risk more, for example, by transferring his bank savings to a stock, if the stock promises higher returns that goes with the incremental risk. One of the important implications of the efficient market hypothesis is that portfolio selection does not make for superior investment results; rather, the achievement of high returns is more properly related to the additional risk than to superior stock-selection ability. Although many fundamental and technical analysts would want to take issue with this claim, still many theoreticians believe that this is correct and would be able to cite empirical research findings to support their views. Expected Value When considering risk, one would want to consider different scenarios about the future and assign probabilities to each one. Normally, companies would thrive when the economy is booming. But there are exceptions. For example, some investments would fare badly during during recessions , and others would thrive under such conditions (such as gold production). It is a good idea, therefore to give weights to alternative possible scenarios and to arrive at a single figure called the Expected Value. Standard deviations Risk for any stock or security is measured by the standard deviation around a given expected value. Where there is wide dispersion of possible outcomes, the standard deviation would be larger, implying more risks. It is common knowledge that the greater the variability of returns, the less certain the actual outcomes would be; so one would prefer less dispersion and variability in the returns for a particular stock. The single figure called the standard deviation provides a clue as to the risk of a particular stock, and stocks could therefore be compared on the basis of its size. For two-asset portfolio of a stock (or projects, as deal with in the attached case), one tries to obtain, firstly the weighted average of the expected returns, and, secondly, the portfolio standard deviation. The two stocks that are combined will interact in such a way that the risk of the two-asset portfolio will be reduced, to depend upon the coefficient of correlation. Where the coefficient of correlation is less than 1, there will be some reduction in the portfolio standard deviation compared to their weighted average. Where the coefficient is negative, it is certain that the portfolio standard deviation will be significantly reduced - and this indicates less risk as a result of the joining of the two assets, stocks, or projects. From the foregoing analysis, one can design an efficient portfolio because the standard deviation has been reduced. A good fund manager can consider many alternative portfolios, each with different expected value and standard deviation. He can build an efficient frontier that represents the the optimal risk-return trade-offs at different risk levels. An efficient frontier is defined as a set of portfolios of investments in which the investor receives maximum return for a given level of risk or a minimum risk for a given level of return (Hirt and Block 2002). The optimal portfolios plotted below along the curve have the highest expected return possible for the given amount of risk. From Investopedia Individual investors can then match their risk-return indifference curves (a concept we learned in microeconomics) with the efficient frontier to find out where they would like to be along this efficient frontier scale. The capital asset pricing model (CAPM) The capital asset pricing model (CAPM) was introduced in the 1960s and 1970s by Sharpe and Lintner as a step forward along the same approach, adding as input the risk-free asset into the analysis. Under this model, the individual may combine an asset yielding a risk-free return with one earning the market rate, and still guaranteeing superior returns on the efficient frontier at all points except one at which they are equal. The capital market line describes demonstrates the trade-off between risk and returns for portfolio managers: Any increase in higher portfolio returns requires a matching level of higher portfolio risks, as indicated by the standard deviation. The security market line shows the same equivalent risk-return trade off pertaining to individual securities. Investors in a stock are deemed rewarded for systematic, market-related risk known as the beta coefficient risk. All other risks - the unsystematic risks that pertain to the peculiar conditions of companies rather than the condition of the general market - can be diversified away through the addition of stocks in the portfolio. Some criticisms of CAPM Although intuitively sound, the capital asset pricing model has not been quite consistent when tested in the real world. For example, an exhaustive study was conducted using the stocks listed in the New York Stock Exchange over a 35- year period. Contrary to the hypothesis that high beta stocks returned higher earnings than low-beta stocks and that low- or zero-beta stocks should yield lower returns, the results were in the reverse. Zero-beta stocks displayed a rate of return that was superior to the risk-free rate and higher beta stocks gave lower returns than anticipated, and the low-beta stocks yielded higher returns than could be predicated from the model. (Black, Jensen & Scholes, cited in Hirt and Block 2002). Another dimension of the theory that has come under criticism has been its apparent unreliability for predicting the future. The method of extrapolating past beta into the future takes the form of a cross-sectional regression between two variables, the broad index on one hand (as independent variable), and the security (as dependent variable) on the other. A regression line is drawn on the basis of the movements of both securities, and over a period of, say, 60, months, a certain figure summarizing that relationship, the beta is obtained. However, because we are not dealing with a diversified portfolio, we know a priori that the price of the security can be subject to extra-systematic influences. Major changes in the industry, shift in popular demand for the companys product, a new product that gains immediate favor with the consumers, and a host of unforeseen events can affect the stocks, events that had no part in the previous calculations based on the model. A stocks beta has very limited uses for predictive purposes. A different but reassuring picture is presented with regard to the betas of diversified portfolios. According to a study by Blume (1975) stability can be achieved with reasonably sized portfolios of even 10 to 20 stocks. Blume measured the correlation between portfolio betas during two different periods and confirmed his hypothesis. At least in defense of the value of the CAPM model, it can be said that large portfolios can have stable betas over time. The model has been proven to be useful in assessing considerations of future market risks and in designing investment strategies, particularly for portfolio managers. Richard Roll (1977) had another issue with the CAPM theory. He said that it is impossible to observe the markets return because of the limited coverage of the existing proxies of the general market. In principle, he said, the market should include all stocks, all financial instruments, and all other types of investment. Indices are imperfect proxies for the true market. Different indices would yield different relationships with stocks and therefore different betas; by changing indices, say from the S&P 500 to the Wilshire 5000, a very different index of stock sensitivity could be obtained. Since the true market has not been known, the CAPM cannot be said to have been properly and fully tested. The alternatives to the CAPM Some realizations of the shortcomings of the CAPM have moved a number of scholars and academicians to search for better or improved alternatives to the model. One alternative was the use of “fundamental” beta, which was basically the use of historical accounting measurements in an equation to produce some kind of valuation model for the investor. While, the author, Barr Rosenberg, has been able to market some of his funds (see Barra.com), there are said to be few converts to his point of view. An alternative model proposed by Stephen Ross (1976), was the the Arbitrage Pricing Theory (APT). This theory has the following assumptions: a) Capital markets are perfectly competitive, b) Investors prefer more wealth to less wealth with certainty, and c) The stochastic process generating asset returns can be represented as a K factor model. Quite simply this model posits less assumptions and is therefore easier to understand and manipulate. Such variables as inflation, growth in the gross domestic product (GDP), major political upheavals, and changes in interest rates, are included in the APT formula. The APT assumes that there are many such factors, and thus it differs from the CAPM in that the latter posits that the only relevant variable is the covariance of the asset with the market portfolio, that is, its beta coefficient. Modern Portfolio Theory I. Part One Analysis Statement of constraints: An investment of £2 million is to be allocated to one or more projects consisting of Project A, B, C, and D. Projects A, C, and D require $1,200,000 each, whilst Project B needs only $800,000. On the basis, only the combinations of Projects A and B, Projects C and B, and Project D and B, would be able to comply with this requirement. Step 1: Calculation of the weighted standard deviations and expected returns of all the projects. Step 2: Calculation of the Portfolio Standard Deviations for the project combinations that met the constraint of $2,000,000 total investment. Step 3: Calculation of the investment ratio for the purpose of weighting and for obtaining an input for the portfolio standard deviation equation. Step 4: Solving for the the standard deviation for 2-asset portfolios. Step. 5: Ranking of the portfolios on the basis of risk-return characteristics. Computing for the Portfolio Standard Deviation (Pσ) of each of these pairs. The formula for the Pσ is as follows: The formula (See Hirt and Block 2002, Brealey et al 1995, Reilly and Brown 1997) Pσ = √ Xa2σa + X2bσb2 + (2)XaXbrab σa σb For the Project A and B portfolio: Pσ = √ (.6)2(8)2 + .(4)2(10)2 + 2(.6)(.4)(.7)(8)(10) = √39.01+28.99 = √ 65.89 Pσ = 8.11 Using the same rather tedious computations, we have obtained the following values of portfolio standard deviation for the two other pairs, Projects B and C, and Projects B and D, as follows: Project B and C = 12.0 Projects B and D = 12.0 Projects A and B together has a low Portfolio Standard Deviation of 8.11, while the other two Portfolios have a similar Portfolio Standard Deviation of 12.0, indicating that the first has a lower risk than the 2nd and 3rd alternatives. This may now be shown in the following graph for risk-reward trade-offs. Conclusion: Project AB has the lowest portfolio standard deviation but does not meet the minimum expected returns of 25 per cent, being only 24.4 (see Table above). Project BD and Project BC have the same level of risk, but Project BD is superior in terms of expected returns with 31.6 per cent. II. Part Two Modern Portfolio Theory: Investment Strategies and Pricing Implications In addition to what has been stated above, the Financial and Investment Dictionary describes modern portfolio theory as different from traditional security analysis by shifting emphasis from analyzing the characteristics of individual investments to the determination of the statistical relationships among the individual securities that comprise the overall portfolio. (Portfolio theory). The portfolio theory approach has four basic steps: a. security valuation-- describing a universe of assets in terms of expected return and expected risk; b. asset allocation decision-- determining how assets are to be distributed among classes of investment, such as stocks or bonds; c. portfolio optimization-- reconciling risk and return in selecting the securities to be included, such as determining which portfolio of stocks offers the best return for a given level of expected risk; and d. performance measurement-- dividing each stocks performance (risk) into market-related (systematic) and industry/security-related (residual) classifications. (ibid) This definition covers the whole range of portfolio theory up to and including the capital asset pricing theory. Asset pricing in modern portfolio theory is based on the calculation of expected returns in relation to expected returns. While different individuals have different risk proclivities, a general rule that one may apply is that the typical risk-averse investor will invest only where the returns justify the risks to be taken. For one who seeks to optimize his results, the efficient frontier is examined to determine which of the investment alternatives are most optimal. The Sharpe ratio is sometimes used as a measure of the amount of incremental return above the risk-free rate a portfolio provides compared to the risk it carries. The portfolio on the efficient frontier with the highest Sharpe Ratio produces a return that is above the efficient frontier, yielding a larger return for a given amount of risk than other risky assets. An efficient investment strategy will ensure the highest return for a given amount of investment, or the least risk (lowest standard deviation) for a given return; or a combination where the risk is reduced at the same time that the expected returns are increased. It is best to combine stocks or projects that are not correlated, or whose coefficient of correlation is negative, for then the combined standard deviation would certainly lower than than the standard deviation each one or of their weighted average. Even when the coefficient correlation is positive, provided it is below +1, some benefits in terms of risk reduction can still be achieved. Based on the CAPM, the unique or unsystematic risk should be diversified away, and only the systematic risk should prevail in order to reduce the risk to a portfolio. It does not mean, however, that all risks are removed, as the systematic risk, the risk affecting all stocks in the economy, is still present. The performance of the portfolio manager is measured in terms of his ability to maximize his risk-adjusted return as well his ability to diversity the portfolio to remove unsystematic risks. This can be achieved not by portfolio selection along, although there should be rational guidelines, but by diversifying the holdings. A portfolio of 20 stocks could achieve this objective. BIBLIOGRAPHY Blume, MF & Friend, I, “A new look at the capital asset pricing model,” Journal of Finance 28, May 1973, pp. 19-33 Brealey, RA, Myers, SC & Marcus, AJ, 1995, Fundamentals of Corporate Finance, McGraw-Hill, Boston, Mass. Efficient frontier, Investopedia, viewed 30 March 2009 at Investopedia http://www.answers.com/efficient%20frontier Hagin, P 1979, Dow-Jones Irwin Guide to modern portfolio theory, Dow Jones-Irwin, Homewood, IL Hirt, GA & Block, SB 2002, Fundamentals of investment management, 7th edn., McGraw Hill, New York Jones, JP 1991, Investments: Analysis and management, 3rd edn., John Wiley & Sons, New York Lintner, J, “The valuation of risk assets and the selection or risky investments in stock portfolios and capital budgets,” Review of Economics and Statistics 47, February 1965, pp. 13-37 Malkiel, BG 1990, A random walk down Wall Street, W. W. Norton, New York Modern portfolio theory, Dictionary of Finance and Investment Terms, Barrons Educational Series, Inc, 2006, viewed 30 March 2009 at http://www.answers.com/topic/modern-portfolio-theory Reilly, FK & Brown, KC 1997, Investment analysis and portfolio management, 5th edn, Dryden Press, Orlando, FL Roll, Richard, “A critique of the asset pricing theory tests,” Journal of Financial Economics 4, no. 4, March 1977, pp. 129-176. _________, “Ambiguity when performance is measured by the securities market line,” Journal of Finance 33, no. 4, September 1978, pp. 1051-1069 Read More